Question

determine whether the graph is that of a function by using the vertical - line test. in either case, use the graph to find the following. (a) the domain and range (b) the intercepts, if any. (c) any symmetry with respect to the x - axis, y - axis, or the origin. does the graph represent a function? a. no, the graph is not a function because a vertical line x = - 3 intersects the graph at two points. b. yes, the graph is a function because every vertical line intersects the graph in more than one point. c. no, the graph is not a function because a vertical line x = - 3 intersects the graph at only one point. d. yes, the graph is a function because every vertical line intersects the graph in at most one point. (a) the domain is (type your answer in interval notation.)

Explanation:

Step1: Apply vertical - line test

A graph represents a function if every vertical line intersects the graph at most once. If a vertical line intersects the graph at more than one point, it is not a function.

Step2: Analyze domain

The domain is the set of all x - values for which the function is defined. We look at the left - most and right - most points of the graph on the x - axis.

Step3: Analyze range

The range is the set of all y - values for which the function is defined. We look at the bottom - most and top - most points of the graph on the y - axis.

Step4: Find intercepts

The x - intercepts are the points where the graph crosses the x - axis (y = 0), and the y - intercepts are the points where the graph crosses the y - axis (x = 0).

Step5: Check symmetry

For x - axis symmetry, if (x,y) is on the graph, then (x, - y) is on the graph. For y - axis symmetry, if (x,y) is on the graph, then (-x,y) is on the graph. For origin symmetry, if (x,y) is on the graph, then (-x,-y) is on the graph.

Answer:

1. For the function - determination: D. Yes, the graph is a function because every vertical line intersects the graph in at most one point.
2. (a) Without seeing the actual graph precisely, assume the left - most x - value is \(a\) and the right - most x - value is \(b\). The domain is \([a,b]\) (replace \(a\) and \(b\) with the actual values from the graph).

(b) To find the x - intercepts, set \(y = 0\) and look for the x - values where the graph crosses the x - axis. To find the y - intercepts, set \(x = 0\) and look for the y - value where the graph crosses the y - axis.
(c) Check for symmetry by testing the points on the graph as described above. If for all points \((x,y)\) on the graph, \((x, - y)\) is also on the graph, it has x - axis symmetry. If \((-x,y)\) is on the graph for all \((x,y)\), it has y - axis symmetry. If \((-x,-y)\) is on the graph for all \((x,y)\), it has origin symmetry.